Test 9 (Lessons 17-18): Families of Functions Solutions

1. $\begin{array}{rcl} y& =& \left|x\right|\end{array}$

Absolute value

2. $y=\sqrt[3]{x}$

Cube root

3. $y=\sqrt{x–3}+2$

Square root

4. $y=–{\left(x+1\right)}^3+2$

Cubic

5. $y=7{\left(x+11\right)}^2$

a = 7, h = –11, k = 0

Sample: The quadratic function stretches vertically by a factor of 7, and shifts left 11 spaces as compared to the parent function$y=x^2.$ There is no vertical movement because k = 0.

6. $y=–\frac{1}{x–8}–15, x\ne 8$

a = –1, h = 8, k = –15

Sample: The rational function reflects over the x-axis, shifts 8 spaces right and down 15 spaces as compared to the parent graph$y=\frac{1}{x}, x\ne 0.$

7. Name the domain and range for the given equation using interval notation. Then name the end behavior.
   $y=–\left|x\right|+3$

$$\begin{gathered} \mathrm{Absolute} \mathrm{value} \\ a=–1, h=0, k=3 \\ \mathrm{domain}: \left(–\infty , \infty \right), \mathrm{range}: \left(–\infty , 3\right] \\ \mathrm{As} x\to +\infty , f\left(x\right)\to –\infty , \mathrm{and} \mathrm{as} x\to \mathit{–}\infty \mathit{,}\mathit{ }f\mathit{\left(}\mathit{x}\mathit{\right)}\to 3 \end{gathered}$$

8. Name the domain and range for the given equation using set-builder notation. Then name the end behavior.
   $y=\sqrt{x+5}–9$

$$\begin{gathered} \mathrm{Square} \mathrm{root} \\ a=1, h=–5, k=–9 \\ \mathrm{domain}: \left\{x|x\in \mathrm{ℝ}, x\ge –5\right\}, \mathrm{range}: \left\{y|y\in \mathrm{ℝ}, y\ge –9\right\} \\ \mathrm{As} x\to +\infty , f\left(x\right)\to +\infty , \mathrm{and} \mathrm{as} x\to –5, f\left(x\right)\to –9 \end{gathered}$$

$$\begin{gathered} a=2, h=6, k=–4 \\ y=2{\left(x–6\right)}^3–4 \\ \mathrm{domain}: \left\{x|x\in \mathrm{ℝ}\right\}, \mathrm{range}: \left\{y|y\in \mathrm{ℝ}\right\} \end{gathered}$$

$$\begin{gathered} a=–1, h=–3, k=2 \\ y=–\frac{1}{x+3}+2 \\ \mathrm{domain}: \left\{x|x\in \mathrm{ℝ}, x\ne –3\right\}, \\ \mathrm{range}: \left\{y|y\in \mathrm{ℝ}, y\ne 2\right\} \end{gathered}$$

11. $y=x$

For the linear parent function, as $x\to +\infty , f\left(x\right)\to +\infty$ and as $x\to –\infty , f\left(x\right)\to –\infty$.

12. $y=x^2$

For the quadratic parent function, as $x\to +\infty , f\left(x\right)\to +\infty$ and as $x\to –\infty , f\left(x\right)\to +\infty .$